Serre's conjecture II

{{unsolved|mathematics|Must the Galois cohomology set of a simply connected semisimple algebraic group over a perfect field of cohomological dimension at most 2 always equal zero?}}

In mathematics, Jean-Pierre Serre conjectured{{Cite journal|last=Serre|first=J-P.|title=Cohomologie galoisienne des groupes algébriques linéaires|year=1962|pages=53–68|journal=Colloque sur la théorie des groupes algébriques}}{{Cite book|last=Serre|first=J-P.|title=Cohomologie galoisienne|publisher=Springer|year=1964|series=Lecture Notes in Mathematics|volume=5}} the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group. Namely, he conjectured that if G is such a group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H¹(F, G) is zero.

A converse of the conjecture holds: if the field F is perfect and if the cohomology set H¹(F, G) is zero for every semisimple simply connected algebraic group G then the p-cohomological dimension of F is at most 2 for every prime p.{{Cite journal|last=Serre|first=Jean-Pierre|date=1995|title=Cohomologie galoisienne : progrès et problèmes|url=http://www.numdam.org/item/?id=SB_1993-1994__36__229_0|journal=Astérisque|volume=227|pages=229–247|mr=1321649|zbl=0837.12003|via=NUMDAM}}

The conjecture holds in the case where F is a local field (such as p-adic field) or a global field with no real embeddings (such as Q({{radic|−1}})). This is a special case of the Kneser–Harder–Chernousov Hasse principle for algebraic groups over global fields. (Note that such fields do indeed have cohomological dimension at most 2.)

The conjecture also holds when F is finitely generated over the complex numbers and has transcendence degree at most 2.{{cite arXiv|last1=de Jong|first1=A.J.|last2=He|first2=Xuhua|last3=Starr|first3=Jason Michael|eprint=0809.5224|title=Families of rationally simply connected varieties over surfaces and torsors for semisimple groups|year=2008|class=math.AG}}

The conjecture is also known to hold for certain groups G. For special linear groups, it is a consequence of the Merkurjev–Suslin theorem.{{cite journal|last1=Merkurjev|first1=A.S.|last2=Suslin|first2=A.A.|title=K-cohomology of Severi-Brauer varieties and the norm-residue homomorphism|journal=Math. USSR Izvestiya|volume=21|year=1983|issue=2|pages=307–340|doi=10.1070/im1983v021n02abeh001793|bibcode=1983IzMat..21..307M}} Building on this result, the conjecture holds if G is a classical group.{{cite journal|last1=Bayer-Fluckiger|first1=E.|last2=Parimala|first2=R.|title=Galois cohomology of the classical groups over fields of cohomological dimension ≤ 2|journal=Inventiones Mathematicae|year=1995|volume=122|issue=1 |pages=195–229 | doi = 10.1007/BF01231443 |bibcode=1995InMat.122..195B|s2cid=124673233|url=http://infoscience.epfl.ch/record/214646 }} The conjecture also holds if G is one of certain kinds of exceptional group.{{cite journal|last=Gille|first=P.|title=Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ≤ 2|journal=Compositio Mathematica|year=2001|volume=125|issue=3|pages=283–325|doi=10.1023/A:1002473132282|doi-broken-date=16 June 2025 |s2cid=124765999|doi-access=free}}

.{{cite book|last=Gille|first=P.|title=Groupes algébriques semi-simples en dimension cohomologique ≤2|series=Lecture Notes in Mathematics|year=2019|volume=2238|doi=10.1007/978-3-030-17272-5|isbn=978-3-030-17271-8 }}